1,438 research outputs found
On vertex coloring without monochromatic triangles
We study a certain relaxation of the classic vertex coloring problem, namely,
a coloring of vertices of undirected, simple graphs, such that there are no
monochromatic triangles. We give the first classification of the problem in
terms of classic and parametrized algorithms. Several computational complexity
results are also presented, which improve on the previous results found in the
literature. We propose the new structural parameter for undirected, simple
graphs -- the triangle-free chromatic number . We bound by
other known structural parameters. We also present two classes of graphs with
interesting coloring properties, that play pivotal role in proving useful
observation about our problem. We give/ask several conjectures/questions
throughout this paper to encourage new research in the area of graph coloring.Comment: Extended abstrac
Problems in extremal graph theory
We consider a variety of problems in extremal graph and set theory.
The {\em chromatic number} of , , is the smallest integer
such that is -colorable.
The {\it square} of , written , is the supergraph of in which also
vertices within distance 2 of each other in are adjacent.
A graph is a {\it minor} of if
can be obtained from a subgraph of by contracting edges.
We show that the upper bound for
conjectured by Wegner (1977) for planar graphs
holds when is a -minor-free graph.
We also show that is equal to the bound
only when contains a complete graph of that order.
One of the central problems of extremal hypergraph theory is
finding the maximum number of edges in a hypergraph
that does not contain a specific forbidden structure.
We consider as a forbidden structure a fixed number of members
that have empty common intersection
as well as small union.
We obtain a sharp upper bound on the size of uniform hypergraphs
that do not contain this structure,
when the number of vertices is sufficiently large.
Our result is strong enough to imply the same sharp upper bound
for several other interesting forbidden structures
such as the so-called strong simplices and clusters.
The {\em -dimensional hypercube}, ,
is the graph whose vertex set is and
whose edge set consists of the vertex pairs
differing in exactly one coordinate.
The generalized Tur\'an problem asks for the maximum number
of edges in a subgraph of a graph that does not contain
a forbidden subgraph .
We consider the Tur\'an problem where is and
is a cycle of length with .
Confirming a conjecture of Erd{\H o}s (1984),
we show that the ratio of the size of such a subgraph of
over the number of edges of is ,
i.e. in the limit this ratio approaches 0
as approaches infinity
Gunrock: GPU Graph Analytics
For large-scale graph analytics on the GPU, the irregularity of data access
and control flow, and the complexity of programming GPUs, have presented two
significant challenges to developing a programmable high-performance graph
library. "Gunrock", our graph-processing system designed specifically for the
GPU, uses a high-level, bulk-synchronous, data-centric abstraction focused on
operations on a vertex or edge frontier. Gunrock achieves a balance between
performance and expressiveness by coupling high performance GPU computing
primitives and optimization strategies with a high-level programming model that
allows programmers to quickly develop new graph primitives with small code size
and minimal GPU programming knowledge. We characterize the performance of
various optimization strategies and evaluate Gunrock's overall performance on
different GPU architectures on a wide range of graph primitives that span from
traversal-based algorithms and ranking algorithms, to triangle counting and
bipartite-graph-based algorithms. The results show that on a single GPU,
Gunrock has on average at least an order of magnitude speedup over Boost and
PowerGraph, comparable performance to the fastest GPU hardwired primitives and
CPU shared-memory graph libraries such as Ligra and Galois, and better
performance than any other GPU high-level graph library.Comment: 52 pages, invited paper to ACM Transactions on Parallel Computing
(TOPC), an extended version of PPoPP'16 paper "Gunrock: A High-Performance
Graph Processing Library on the GPU
Local Conflict Coloring
Locally finding a solution to symmetry-breaking tasks such as
vertex-coloring, edge-coloring, maximal matching, maximal independent set,
etc., is a long-standing challenge in distributed network computing. More
recently, it has also become a challenge in the framework of centralized local
computation. We introduce conflict coloring as a general symmetry-breaking task
that includes all the aforementioned tasks as specific instantiations ---
conflict coloring includes all locally checkable labeling tasks from
[Naor\&Stockmeyer, STOC 1993]. Conflict coloring is characterized by two
parameters and , where the former measures the amount of freedom given
to the nodes for selecting their colors, and the latter measures the number of
constraints which colors of adjacent nodes are subject to.We show that, in the
standard LOCAL model for distributed network computing, if l/d \textgreater{}
\Delta, then conflict coloring can be solved in rounds in -node graphs with maximum degree
, where ignores the polylog factors in . The
dependency in~ is optimal, as a consequence of the lower
bound by [Linial, SIAM J. Comp. 1992] for -coloring. An important
special case of our result is a significant improvement over the best known
algorithm for distributed -coloring due to [Barenboim, PODC 2015],
which required rounds. Improvements for other
variants of coloring, including -list-coloring,
-edge-coloring, -coloring, etc., also follow from our general
result on conflict coloring. Likewise, in the framework of centralized local
computation algorithms (LCAs), our general result yields an LCA which requires
a smaller number of probes than the previously best known algorithm for
vertex-coloring, and works for a wide range of coloring problems
Streaming and Massively Parallel Algorithms for Edge Coloring
A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. (Note that the maximum degree, Delta, is a trivial lower bound.) In this paper, we revisit this fundamental problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model:
- Massively Parallel Computation: We give a randomized MPC algorithm that with high probability returns a Delta+O~(Delta^(3/4)) edge coloring in O(1) rounds using O(n) space per machine and O(m) total space. The space per machine can also be further improved to n^(1-Omega(1)) if Delta = n^Omega(1). Our algorithm improves upon a previous result of Harvey et al. [SPAA 2018].
- Graph Streaming: Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors.
We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors with high probability if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors
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